Zettelkasten/Fleeting Notes/Class/Bayesian Signal Processing/variance-and-convergence.md

Variance and Convergence

Definition of Variance

\text{Var}(X) = E\{(x - \mu)^2\}

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Variance of the Monte Carlo Estimator

\text{Var}(\hat{I}) = \frac{(b-a)^2}{n} \text{Var}(f(x))

Convergence Rate

Standard deviation scales as:

\sigma \sim \frac{1}{\sqrt{n}}

This is good for a small number of dimensions.

However: With higher dimensionality, variance gets exponentially worse (curse of dimensionality).

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Multi-Dimensional Case

Combine with Monte Carlo Integration techniques (importance sampling, stratified sampling, etc.) to manage variance in high dimensions.